Primal and dual convergence of a proximal point exponential penalty method for linear programming

We consider the diagonal inexact proximal point iteration u(k) - u(k-1)/lambda(k) is an element of - partial derivative(epsilonk)f(u(k), r(k)) + v(k) where f(x, r) = c(T)x + rSigmaexp[(A(i)x - b(i))/r] is the exponential penalty approximation of the linear program min{c(T)x : Ax less than or equal to b}. We prove that under an appropriate choice of the sequences lambda(k), epsilon(k) and with some control on the residual v(k), for every r(k) --> 0(+) the sequence u(k) converges towards an optimal point u(infinity) of the linear program. We also study the convergence of the associated dual sequence mu(i)(k) = exp[(A(i)u(k) - b(i))/r(k)] towards a dual optimal solution.